Question

In a Keynesian model of a closed economy, if autonomous consumption is R100 million, investment spending is R200 million, government spending is R300 million, the marginal propensity to consume is 0.8 and the tax rate is 0.25 (or 25%), and full employment level of income is R2 000 million, by how much should government spending increase to achieve full employment?

Answer 1

To determine how much government spending should increase to achieve full employment in a Keynesian model, we first need to calculate the initial equilibrium income using the given data.

Given Data:

• Autonomous Consumption (C₀): R100 million
• Investment (I): R200 million
• Government Spending (G): R300 million
• Marginal Propensity to Consume (MPC): 0.8
• Tax Rate (t): 0.25
• Full Employment Level of Income (Yₑ): R2,000 million

Step 1: Calculate the Initial Equilibrium Income

In a Keynesian model, the equilibrium income (Y) can be expressed as:

\[ Y = C + I + G \]

Where C (consumption) is influenced by income and the tax rate. The consumption function in this context takes taxes into account:

\[ C = C_0 + MPC \times (Y - T) \]

Where T (total taxes) can be calculated as:

\[ T = t \times Y = 0.25 \times Y \]

Substituting T in the consumption function:

\[ C = C_0 + MPC \times (Y - 0.25Y) = C_0 + MPC \times (0.75Y) \]

Now, substituting the values we have:

\[ C = 100 + 0.8 \times 0.75Y = 100 + 0.6Y \]

Step 2: Set Up the Equation for Y

Now we can set up the equilibrium equation:

\[ Y = C + I + G = (100 + 0.6Y) + 200 + 300 \]

This simplifies to:

\[ Y = 600 + 0.6Y \]

Step 3: Solve for Y

To isolate Y, we move the \(0.6Y\) to the left:

\[ Y - 0.6Y = 600 \] \[ 0.4Y = 600 \] \[ Y = \frac{600}{0.4} = 1500 \text{ million} \]

Thus, the initial equilibrium income is R1,500 million.

Step 4: Calculate the Change Needed to Reach Full Employment

To achieve full employment at Yₑ = R2,000 million, we need to find the required change in output (ΔY):

\[ \Delta Y = Y_e - Y = 2000 - 1500 = 500 \text{ million} \]

Step 5: Calculate the Multiplier Effect

The spending multiplier (k) is calculated as:

\[ k = \frac{1}{1 - MPC(1 - t)} = \frac{1}{1 - 0.8(1 - 0.25)} = \frac{1}{1 - 0.8 \times 0.75} = \frac{1}{1 - 0.6} = \frac{1}{0.4} = 2.5 \]

Step 6: Calculate the Required Increase in Government Spending

To find the necessary increase in government spending (ΔG) for the required increase in income (ΔY), we use:

\[ \Delta Y = k \times \Delta G \]

Rearranging gives:

\[ \Delta G = \frac{\Delta Y}{k} \] \[ \Delta G = \frac{500}{2.5} = 200 \text{ million} \]

Conclusion

To achieve full employment, government spending should increase by R200 million.